Potential vorticity, first introduced by Ertel (1942) as dry potential vorticity, is a fundamental concept of atmospheric dynamics. Because of its conservation following fluid motion in a frictionless and adiabatic flow and its invertibility in a balanced system, the dry potential vorticity is very useful in both diagnostic and prognostic studies of the atmosphere.
When latent heat of condensation is taken into account, the dry potential vorticity is not conserved. However, by replacing the potential temperature with the equivalent potential temperature, it is possible to define the moist potential vorticity. The moist potential vorticity becomes conserved in moist adiabatic processes; it has been extensively used in studies of conditional symmetric instability in baroclinic systems since first proposed as a possible mechanism for the formation of frontal rainbands by Bennetts and Hoskins (1979) and Emanuel (1979, 1983). These authors showed that a negative moist potential vorticity is a sufficient condition for two-dimensional frictionless conditional symmetric instability. Recently, Cao and Cho (1995) and Persson (1995) have investigated the generation of moist potential vorticity in extratropical cyclones.
In the cloudy systems associated with extratropical cyclones, negative Celsius temperatures and ice are usual. Moist potential vorticity is not conserved when latent heat of freezing is taken into account. Therefore, if we wish to investigate the possible effect of ice in the generation of potential vorticity, we have to define a potential vorticity that takes into account the possibility of latent heat of freezing release.
In this paper we propose a generalized definition of potential vorticity and the possible effect of ice as a source of generalized potential vorticity is investigated.
2. Generalized potential vorticity
Since the dry air mass is greater than other component masses, it could be assumed that the dry air guides the motion of the system; that is, the center of mass velocity is equal to the dry air velocity. Under this assumption, changes of entropy temperature are caused just by entropy changes (Hauf and Holler 1987). Consequently, if reversibility is assumed, dθs/dt = 0 and θs remains constant for any parcel of cloudy air.
3. Effect of ice on the generation of generalized potential vorticity
The situation schematized in Fig. 1 may be produced in extratropical cyclones, where cloud bands are nearly parallels to the fronts. On the other hand, the ice content decreases from the inner zones to the boundaries of the ice regions in cloud bands. It seems clear that the vector ∇(i/T) points in opposite directions in the two sides of the ice region that are nearly parallels to the front. For example, if cloud bands associated with a north–south oriented cold front are considered, Fig. 1b could represent the situation on the east side of ice region and Fig. 1a on the west side.
A reasonable value for α2 is 0.1° (Bennetts and Hoskins 1979). This implies a rate of about 0.4–0.5 PVU/day for (dqg/dt)m. This magnitude is consistent with the values shown in previous works (Bennetts and Hoskins 1979; Cao and Cho 1995; Persson 1995). If we choose the same value for α1 as for α2, the ice solenoid term is about 0.04 PVU/day. If α1 = 1° is appropriate, the ice solenoid term contribution is about 0.4 PVU/day. Considering that a change of 0.3 PVU/day would be the minimum rate detectable in observations, the value for α1 should be at least 0.7°. Therefore, an observable contribution of spatial gradients of ice concentration on changes of generalized potential vorticity requires an angle between θ and i/T surfaces about an order of magnitude higher than the angle between θ and θe surfaces. In consequence, the contribution of ice solenoid term would probably be smaller than the classic solenoid term contribution. However, because the spatial distribution of ice concentration is more irregular than the water vapor distribution, there will be cases where α1 is considerably higher than α2, especially for the 10-km scale we are considering. In these cases, the ice solenoid term should be taken into account.
The possible effect of ice on changes of potential vorticity is considered. For this purpose we define a “generalized” potential vorticity using Hauf and Holler’s entropy temperature. We find that two terms contribute (if the flow is assumed reversible and frictionless) to change the generalized potential vorticity: the classic solenoid term and a new term that involves the spatial gradient of existing ice concentration. An estimate of this “ice solenoid” term shows that an observable contribution of ice on changes of generalized potential vorticity requires an angle between θ and i/T surfaces about an order of magnitude higher than the angle between θ and θe surfaces. So the contribution of the ice solenoid term would probably be smaller than the classic solenoid term in most of the cases.
This paper has been partially supported by CICYT (AMB-94-0701).
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